| bgcolor="#BCD2EE" | Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8-class groups predicted by the Cohen-Lenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8-class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8-class ranks in most of these small families is given by the Cohen-Lenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4-Selmer groups in the quadratic twist family of an elliptic curve with full 2-torsion.

Revision as of 20:37, 19 September 2016

Contents

Sep 8

Arunabha Biswas

Limiting values of higher Mahler Measure and cyclotomic polynomials.

We consider the k-higher Mahler measure m_k(P) of a Laurent polynomial P as the integral of log^k |P| over the complex unit circle. In number theory, Lehmer's conjecture and the appearance of higher Mahler measures in L-functions are the main sources of motivation for studying various properties of m_k(P). Beyond number theory, Mahler measure has connections with topological entropies of dynamical systems and polynomial knot invariants. In this talk I shall present (1) an explicit formula for the value of |m_k(P)|/k! as k approaches infinity, (2) some asymptotic results regarding m_k(P) and (3) a scheme to approximate special values of a class of L-functions.

Sep 15

Naser T. Sardari

Discrete Log problem for the algebraic group PGL_2.

We consider the problem of finding the shortest path between a given pair of vertices in the LPS Ramanujan graphs $X_{p,q}$ where $p$ is a fixed prime number and $q$ is an integer. We give a polynomial time algorithm in $\log(q)$ which returns the shortest path between two diagonal vertices under a standard conjecture on the distribution of integers representable as sum of two squares and assuming one can factor quickly. Numerically, for a typical pair of vertices corresponded to diagonal elements the minimal path has a length about $3\log(q)+ O(1)$ while provably, there are pairs of points with distance at least $4\log(q)+ O(1)$ . For a general pair of vertices, we write it as a product of three Euler angels and as a result for a typical pair we find a path with distance $9 \log(q)+ O(1)$.

Sep 22

Alex Smith

Statistics for 8-class groups and 4-Selmer groups

Assuming the grand Riemann hypothesis, we verify that the set of quadratic imaginary fields has the distribution of 8-class groups predicted by the Cohen-Lenstra heuristic. To do this we prove that, in families of quadratic fields parameterized by a single prime p, the 8-class rank is determined by the Artin symbol of p in a certain extension of the rationals. Using Chebotarev's density theorem, we find that the distribution of 8-class ranks in most of these small families is given by the Cohen-Lenstra heuristic. We can bundle these small families together to get the full result, with GRH necessary to control error bounds in this process. By analogous means, we also find the distribution of 4-Selmer groups in the quadratic twist family of an elliptic curve with full 2-torsion.

Sep 29

Steve Lester

title coming soon

abstract coming soon

Oct 6

Since the mid-1980s, when the study of arboreal Galois representations first began, most results have concerned the representations induced by quadratic rational maps. In the higher degree case, by contrast, very little has been known. I will discuss some recent results pertaining to higher degree polynomials over number fields. This will include a partial solution to a conjecture made by R.W.K. Odoni in 1985.